Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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7
votes
1answer
324 views

Can A be singular? [duplicate]

Let $A\in \mathbb{C}^{n\times n}$ satisfy $$A^{2}+A+I=0 $$ Can A be singular? So I have: $$ (A-I)(A^{2}+A+I)=0\\ A^{3} = I \\ (\det A^{3}) = \det(I) \\ (\det A)^{3} = 1\\ \det A\neq 0 $$ So $A$ is ...
19
votes
8answers
3k views

Why do negative exponents work the way they do? [closed]

Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent? $$a^{-b} = \frac{1}{a^b}$$
5
votes
0answers
75 views

Eigenvectors of difference of inverse matrices

I have two matrices $A$ and $B$, symmetric and positive semi-definite (in fact, they are covariance matrices), and I am interested in computing the eigenvectors of the matrix $A^{-1}-B^{-1}$. From ...
0
votes
1answer
32 views

If an analytic function $f: \mathbb{R}^2 \to \mathbb{R}^2$ is locally invertible at $(x_0, y_0)$, then $Df(x_0,y_0) \not = 0$.

I am trying to show that if an analytic function $f: \mathbb{R}^2 \to \mathbb{R}^2$ (i.e. $f$ satisfies the Cauchy-Riemann equations) is locally invertible at $(x_0, y_0)$, then $Df(x_0,y_0) \not = ...
0
votes
1answer
17 views

Find a right inverse of a map with gauss brackets.

I am having a composition of two maps: $$ f:\mathbb{R}->\mathbb{R_0^+},f(x)=x^2 $$ $$ g:\mathbb{R_0^+}->\mathbb{\mathbb{N}},g(x)=\lfloor x\rfloor $$ $$h=g\circ f:\mathbb{R}->\mathbb{N_0}$$ ...
2
votes
7answers
644 views

How come the function and the inverse of the function are the same?

What is the inverse of the function: $$f(x)=\frac{x+2}{5x-1}$$ ? Answer: $$f^{-1}(x)=\frac{x+2}{5x-1}$$ Can one of you explain how the inverse is the same exact thing as the original equation?
3
votes
1answer
33 views

When is Block-Partitioned Matrix Invertible?

Suppose I have a block partitioned matrix \begin{equation} \begin{bmatrix} \mathbf{X}_1^{\top}\mathbf{X}_1 & \mathbf{X}_1^{\top}\mathbf{X}_2 \\ \mathbf{X}_2^{\top}\mathbf{X}_1 & ...
0
votes
1answer
21 views

What is a quickest way to find inverses of functions of two variables?

Suppose I have $X_1 = aY_1+bY_2$ $X_2 = cY_1+dY_2$ How is the quickest (or most efficient way ) to find the inverse functions? The current way I am doing it is attempting to solve for Y1 in the ...
3
votes
1answer
56 views

Explicit formula for inverse matrix elements

Let $A$ be an $n \times n$ invertible matrix with \begin{align} \left(\begin{array}{ccc} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} ...
0
votes
3answers
30 views

Multiplicative inverse

What is the multiplicative inverse of 7 modulo 11? Is this correct: $$7 = 11(0) +7$$ $$11 = 7(1) +4$$ $$7 = 4(1) +3$$ $$4 = 3(1) +1$$ We then take 3 equations: $$4 = 11 + 7(-1)$$ $$3 = 7 + 4(-1)$$ ...
1
vote
0answers
30 views

Finding the inverse of an integral

I'm looking for a computational approach here, since I don't think there is a closed-form solution. I have the following: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
0
votes
0answers
11 views

Sum of Inverse Partitioned Matrix

Given a matrix $X(n\times p)$, divide $X$ by row into $K$ parts, $X_1, X_2...X_K$ each of which consists of the same amount of row vectors in $X$ as its own row vectors. Now consider ...
5
votes
1answer
142 views

Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is such a diagonal ...
2
votes
4answers
71 views

Inverse Matrices and Infinite Series

Given that $C=I+A+A^2+A^3+ \ldots$ Prove that I-A is the inverse of $C$ Hint: Use the infinite series technique for finding inverse of a matrix. Now I know with an infinite geometric series with a ...
1
vote
1answer
27 views

Using Derivatives and Tangent Line to Find Area

Let $(a, b)$ be an arbitrary point on the graph of $y=\frac1x$ ($x>0$). Prove that the area of the triangle formed by the tangent line through $(a,b)$ and the coordinate axes is $2$ square units. ...
0
votes
1answer
65 views

Need help to find the inverse of a mapping implicitly

Let $f: \mathbb R^3\to \mathbb R^3$ be the linear mapping which reflects $x$ over the plane $x1+x2+x3=0$. You are given that the standard matrix for $f$ is: ...
1
vote
0answers
25 views

What's the algebraic definition of the inverse of a function?

I have a function $f(x)$, using the logic it's relatively easy to formally define what the inverse should be like, relatively to domain and codomain elements, especially using the surjective and ...
3
votes
1answer
95 views

checking whether functions satisfy Inverse Function Theorem.

I've my exam tomorrow and this question is expected to come but donot know how to solve... Here's the INVERSE FUNCTION THEOREM stated in my notes: It says: Let $E\subseteq \mathbb R^n$ be open ...
1
vote
1answer
122 views

A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding $A^{-1}$

The question is: A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding A^-1. I have looked at other similar questions on this site: 1. Here 2. and Here But they use ...
0
votes
1answer
26 views

Finding hermitian conjugate and inverse of a complex matrix

I have the following matrix: $$ F = [e^{i\frac{2\pi kl}{n}}]^{n-1}_{k,l=0} \in \mathbb{C}^{n,n} $$ for $n = 1,2,3,...,i$ I need to find $F^HF$ and $F^{-1}$ where $F^H$ is a hermitian conjugate ...
3
votes
3answers
101 views

What is the inverse of $f(x)=\frac{e^x+e^{-x}}{e^x-e^{-x}}$?

please help me to find out the inverse this function, $$f(x)=\frac{e^x+e^{-x}}{e^x-e^{-x}}$$ I know that, let $$y=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$ and if I find $x=\cdots$ then that is the ...
4
votes
1answer
112 views

If $f(x) = \sum \limits_{n=0}^{\infty} \frac{x^n}{2^{n(n-1)/2} n!}$ then $f^{-1}(f(x)-f(x-1))-\frac{x}{2}$ is bounded

For every $x>0$, let $$f(x) = \sum \limits_{n=0}^{\infty} \dfrac{x^n}{2^{n(n-1)/2} n!}.$$ Let $f^{-1}$ be the functional inverse of $f$. How to show there exists a positive real constant $C$ such ...
4
votes
5answers
45 views

Tool for expressing $x=f^{-1}(y)$ if $y=f(x)$ is given

I have many equations of nature - $y=ax^{12}+bx^5+5x^4+1$ For all these equations, I need to express x in terms of y. What tool or software would you recommend for this? I would much prefer to ...
0
votes
0answers
32 views

Inverse Laplace transform of $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$ [duplicate]

I have been desperately trying to find the inverse laplace transform using the complex inversion formula for this question. $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$ I have found it extremely ...
2
votes
1answer
39 views

Calculate the inverse of $h(x)=f(2x)$

I have to calculate the inverse $f^{-1}(x)$ of $y=f(x)=2x-1$ and it is simple for this kinds of functions Let $x=f(y)=2y-1$ $x+1=2y$ $\displaystyle\frac{x+1}{2}=y$ We now have the inverse ...
0
votes
0answers
30 views

Determine matrix from linear transformation

Let $T_{1}$ and $T_{2}$ be linear transformations given by $$T_{1}([x_{1}, x_{2}])=[3x_{1}+5x_{2}, 4x_{1}+7x_{2}]$$ $$T_{2}([x_{1}, x_{2}])=[2x_{1}+9x_{2}, x_{1}+5x_{2}]$$ Find a matrix A such that ...
0
votes
1answer
28 views

finding the inverse of a matrx

In order to decrypt a cipher text using hill cipher, we must first find the inverse matrix of a given matrix. From this link http://en.wikipedia.org/wiki/Hill_cipher, ...
1
vote
0answers
13 views

Speed of pseudo-inverse (with possibly ill-conditioned matrices)

I am computing the pseudo-inverse of several matrices of identical size $m \times n$ . However, computation (e.g. with the LAPACK pinv) seems to be much slower in some cases (5 to 10 times slower). ...
0
votes
1answer
27 views

Inverse Laplace transformation of (s^2-4s-2)/((s^2+2)^2)

I approached this problem as follow: $1.$ rewrote $(s^2-4s-2)$ into $(s-2)^2-6$ $2.$ Now break the function into 2 parts: $\frac{(s-2)^2}{(s^2+2)^2} + \frac{6}{(s^2+2)^2}$ the Laplace inverse ...
0
votes
0answers
43 views

What is the inverse of $f(x)=x^{x^x}$?

I'm curious to find the inverse of $ f(x)=x^{x^x} $ As an added extra, I'm already familiar with the Lambert Product Log function.
3
votes
1answer
126 views

Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
0
votes
2answers
48 views

How to show that $AX=B$ has unique solution for invertible matrix $A$

If $A$ is an invertible $n \times n$ matrix, show that $AX=B$ has a unique solution for any $n \times k$ matrix $B$. I'm not sure where to start. What I have is that, if $A$ is invertible then ...
5
votes
2answers
94 views

Integral involving inverse of $x^x$

My brother gave me the following problem: Let $f:[1;\infty)\to[1;\infty)$ be such that for $x≥1$ we have $f(x)=y$ where $y$ is the unique solution of $y^y=x$. Then calculate: $$ \int_0^e f(e^x)dx $$ ...
1
vote
1answer
31 views

Schwarzian derivative of inverse function.

Let $\mathcal{D}$ denote the Schwarzian derivative. I have to prove that if $\mathcal{D}f(x)$ exists $\forall x$ then $\mathcal{D}f^{-1}$ exists $\forall x\in D_{f^{-1}}$ then find a formula. I tried ...
1
vote
1answer
55 views

Proof that if $A$ is similar to $B$, then $B$ is similar to $A$

$A$ is similar to $B$ if there is an invertible matrix $S$ such that $B = S^{-1}AS$. Prove that if $A$ is similar to $B$, then $B$ is similar to $A$. So if $A$ is similar to $B$ then $B = ...
1
vote
2answers
56 views

The inverse of AR structure correlation matrix / Kac-Murdock-Szeg ̈o matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots ...
1
vote
3answers
46 views

show that every rational number has one and only one multiplicative inverse

I am stumped and have no idea on how I prove this. I don't know what else to say. I am beyond lost.
2
votes
1answer
48 views

Circle Equation Surjectivity

Consider the circular function $g:\mathbb{R}^{2} \to \mathbb{R}^{+}$, $g(x,y)=x^{2}+y^{2}$. Show that it is surjective and continuous. Note This post has been amended in accordance with the ...
0
votes
1answer
33 views

Inverse matrices properties.

I know about the properties of matrix multiplication for multiplication such as $A(BC)=(AB)C$. However I'm curious if $(AC)B$ would also have the same value. I'm asked to represent $A$ in terms of $B$ ...
0
votes
2answers
47 views

Showing there is no invertible function $f: \mathbb{R} \to \mathbb{R}$

I'm wondering whether there is an invertible function $f: \mathbb{R} \to \mathbb{R}$ such that $f(-1)=0$, $f(0)=1$ and $f(1)=-1$. I think it's not but I'm missing a real proof. The easiest would be ...
0
votes
0answers
9 views

Sherman Morrison Formula for hermitian updates

I have a problem in which, in principle I can apply twice Sherman-Morrison formula but it seems to me that for this case, there should be a simpler solution so my question is "May the process ...
2
votes
1answer
44 views

How to get tangent of inverse of curve??

Ok so my question is. Let $ f(x)=(1/7)x^3+21x-1.$ and let y=g(x) be the inverse function of f. Determine all points on the graph of the inverse function g so that the tangent line is perpendicular to ...
0
votes
1answer
23 views

What will $A^+A$ and $A^gA$ actually or exactly get if $A$ is not invertible?

I know if $A$ is invertible then $A^{-1}$ is the inverse of $A$, and $AA^{-1}=A^{-1}A=I$. I just learnt the concept of Generalized inverses and Moore–Penrose pseudoinverse. For a matrix $A$ that is ...
0
votes
1answer
18 views

conditions for Gauss_jordan elimination with no pivoting

Please note that here is Gauss_jordan elimination which help us get inverse of A. I am wondering, is there any condition that it could work without pivoting? I try to prove this under column ...
0
votes
2answers
17 views

Help with proving matrix transpose and inverses.

I am really struggling with these type of proofs. Could someone please give me hints on how to prove them, I do know the basic properties of transpose and inverse. If $ \mathbf{A} $ is invertible and ...
0
votes
1answer
22 views

Quadratic Equation with Matrix [Prove Invertible]

The problem is: "The $2\times 2$ matrix A satisfies $A^2-4A-7I=0,$ where I is the $2\times 2$ identity matrix. Prove that A is invertible." The hint given is: "We are trying to a matrix that is ...
0
votes
2answers
29 views

How does one compute the inverse of the function $f$ that satisfies $f(3x-2) = x-1$? [closed]

The problem is: Given $f: \mathbb{R} \to \mathbb{R}$ such that $f(3x-2) = x-1$, find $f^{-1}(x)$. It would be great if you could help me on this one
0
votes
0answers
34 views

The converse of the inverse function theorem

The inverse function theorem: A continuously differentiable function $F=(F₁,...,F_{r+1})$ defined from an open set $U⊂ℝ^{r+1}$ into $ℝ^{r+1}$ is invertible at a point $z=(s₁,s₂,...,s_{r},s_{r+1})∈U$ ...
0
votes
0answers
15 views

Inverse of a 2x2 principal submatrix whose inverse is known

Let $H$ be a $n\times n$ symmetric positive definite matrix. What is the (computationally) quickest way to obtain $H_{ij}$, the $2\times 2$ matrix whose inverse is the principal submatrix of the ...
1
vote
1answer
52 views

Inverse of the sum of two orthogonal projections

I am trying to find out, if there is a formula for finding the inverse of the sum of two orthogonal projections. So basically my questions is: If $\left[\mathbf{A},\mathbf{B}\right]$ is full rank, ...